#150651

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	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

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

normal subgroups only

This group has 284 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18,21,27,23,29,31,47)(19,30,20,25,33,42,34,46)(22,26,38,44,39,28,37,40)(24,35,41,32,43,36,45,48);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19,22,32,23,33,39,48)(18,24,28,42,29,43,26,30)(20,31,35,38,34,21,36,37)(25,44,45,47,46,40,41,27);(2,7)(6,14)(8,15)(9,16)(17,20)(18,25)(19,31)(21,33)(22,35)(23,34)(24,40)(26,41)(27,42)(28,45)(29,46)(30,47)(32,38)(36,39)(37,48)(43,44)

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=gh,e^2=i,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ch,d^b=dg,e^b=efghi,d^c=di,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, 57847);