#150404

order	512
nilpotency class	5
isoclinism	isoclinic class of #150397
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#391	C₂³
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ #2310 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

421 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(2,5,6,7)(9,10,13,19,15,21,32,50)(11,22,24,49,26,41,54,66)(12,17,28,45,30,47,58,69)(14,18,36,48,37,20,33,51)(16,38,40,25,42,31,61,34)(23,43,52,67,53,63,57,71)(27,39,56,64,55,65,62,72)(29,44,60,68,35,46,59,70);(1,2)(3,7)(4,6)(5,8)(9,11,15,26)(10,16,21,42)(12,27,30,55)(13,31,32,38)(14,34,37,25)(17,43,47,63)(18,41,20,22)(19,49,50,66)(23,35,53,29)(24,33,54,36)(28,57,58,52)(39,44,65,46)(40,51,61,48)(45,64,69,72)(56,59,62,60)(67,70,71,68);(9,12,15,30)(10,17,21,47)(11,23,26,53)(13,28,32,58)(14,35,37,29)(16,39,42,65)(18,46,20,44)(19,45,50,69)(22,43,41,63)(24,52,54,57)(25,55,34,27)(31,62,38,56)(33,60,36,59)(40,64,61,72)(48,70,51,68)(49,67,66,71)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=g,e^2=i,f^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=efgi,f^b=fi,d^c=dh,f^c=fi,e^d=ehi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57600);