#151885

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

384 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,f^2=g,g^2=h,h^2=i,b^a=bd,c^b=ci

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286359, #72286363, #72286704, …
maximal normal subgroup of	#72286359, #72286363, #72286704, …
maximal quotient of	#72286396, #72321570

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59081);