#128206

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₂×C₈² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₄²
commutator	C₄	C₂×C₈²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#97
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

342 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(17,18,22,28,23,30,38,45,24,29,37,44,21,27,20,26)(19,31,34,47,35,25,32,41,36,43,39,40,33,42,48,46);(2,5)(6,8)(9,10,11,13,12,14,15,16)(17,19)(18,25,29,42)(20,32)(21,33)(22,39)(23,35)(24,36)(26,40,45,47)(27,31,30,43)(28,46,44,41)(34,37)(38,48);(1,3,4,7)(2,6,5,8)(17,20,21,37,24,38,23,22)(18,26,27,44,29,45,30,28)(19,32,33,34,36,48,35,39)(25,40,31,41,42,47,43,46)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=d,d^2=i,e^2=g,f^2=h,g^2=i,b^a=bd,f^a=fi,e^b=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210602, #72210635, #72210682
maximal normal subgroup of	#72210602, #72210635, #72210682
maximal quotient of	#72210645, #72210687, #72234227

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35402);