#92987

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×C₂²⋊C₈	C₂²
socle	C₂³	#323
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 448 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)(7,11,12,9)(8,10)(13,14,17,22,18,23,32,40,19,24,33,41,16,21,30,38)(15,25,27,42,26,36,43,39,28,37,31,44,29,20,34,35);(1,2,3,4)(6,9)(10,11)(13,15)(14,20)(16,29)(17,31)(18,26)(19,28)(21,37)(22,39)(23,25)(24,36)(27,33)(30,43)(32,34)(35,41)(38,42)(40,44)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=g,g^2=i,b^a=bc,c^a=cf,f^a=fh,c^b=ci,d^b=dfi,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, 183);