#7701852

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4966 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=i,e^2=g,f^2=gh,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7609048);