#149499

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

Aliases

No aliases for this group.

Subgroups

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

212 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:

(1,2,3,6,4,5,7,8);(2,5)(6,8)(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);(10,15)(16,25)(17,26)(18,27)(28,35)(29,36)(30,37)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225137, #72225138, #72225517
maximal normal subgroup of	#72225137, #72225138, #72225517
maximal quotient of	#72225137

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56695);