#93072

order	512
nilpotency class	4
isoclinism	isoclinic class of #7092
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#331
commutator	C₄×C₈	C₄²
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1296
prime radical	C₂³	#332
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

153 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 153 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 256

A minimal faithful permutation representation — 2 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=i,e^2=fghi,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268);