#494197

order	512
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂³	C₂²×D₈
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=fh,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320179, #72320205, #72320231, …
maximal normal subgroup of	#72320179, #72320205, #72320231, …
maximal quotient of	#72320480

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401393);