#124632

order	512
nilpotency class	4
isoclinism	isoclinic class of #11836
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1514
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×#97	C₂³
socle	C₂	#7603
prime radical	#3147	C₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

333 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 333 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31828);