#102255

order	512
nilpotency class	3
isoclinism	isoclinic class of #102154
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

438 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=egi,d^2=gh,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,f^b=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9451);