#7701094

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#46367	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5020 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262649, #72316538
maximal normal subgroup of	#72262649, #72316538
maximal quotient of	#72262656, #72316492

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7608290);