#96258

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

854 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 854 conjugacy classes of subgroups — too many to draw. Only its 250 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) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=hi,d^2=g,f^2=i,b^a=bd,d^a=dg,f^a=fgh,c^b=chi,d^b=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217725
maximal normal subgroup of	#72217725
maximal quotient of	#72213034

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3454);