#146962

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

710 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322932, #72322951, #72322958
maximal normal subgroup of	#72322932, #72322951, #72322958

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54158);