#92991

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#99
commutator	C₂²×C₄	C₄×C₈
Frattini, cosocle	C₂²×C₄⋊C₈	C₂²
socle	C₂³	#324
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂²×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

318 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=hi,d^2=g,g^2=i,b^a=bc,c^a=cf,e^a=eh,f^a=fh,c^b=ci,d^b=dfhi,d^c=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 187);