#7640258

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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

4957 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)(10,14)(11,12)(17,18,21,22)(19,25,27,23)(20,28,29,24)(26,30,32,31);(2,6)(7,8)(9,10)(11,16)(12,15)(13,14)(18,22)(20,29)(23,25)(26,32);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,15,14,16)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31);(19,27)(23,25)(26,32)(30,31);(1,4,5,3)(2,8,6,7)

Polycyclic presentation:

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

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, 7547454);