#495485

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄×C₈	C₂³
commutator	C₂²	C₂³×C₁₆
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂³	C₂³×C₈
prime radical	C₂²×#41	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1050 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322698, #72322855, #72323010, …
maximal normal subgroup of	#72322698, #72322855, #72323010, …
maximal quotient of	#72322704

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402681);