#7638240

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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

4793 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,e^c=ehi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7545436);