#150709

order	512
nilpotency class	5
isoclinism	isoclinic class of #150594
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

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	C₂³	C₂×#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄² ◁ #2327 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

241 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 241 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(2,6)(3,4)(7,8)(9,10,13,19,15,21,28,37)(11,22,23,36,12,17,25,34)(14,18,30,35,31,20,29,38)(16,27,32,26,33,39,40,24);(1,2)(3,6)(4,8)(5,7)(9,11,15,12)(10,16)(13,27,28,39)(14,26,31,24)(17,20)(18,22)(19,34)(21,33)(23,29,25,30)(32,35)(36,37)(38,40);(1,3,5,4)(2,6,7,8)(9,12,15,11)(10,17,21,22)(13,23,28,25)(14,26,31,24)(16,18,33,20)(19,36,37,34)(27,29,39,30)(32,38,40,35)

Polycyclic presentation:

2^9|a^2=e,b^2=hi,c^2=gh,d^2=gi,e^2=i,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=ch,d^b=dg,e^b=efg,f^b=fi,d^c=di,e^c=ei,e^d=ehi,f^e=fi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57905);