#7702272

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
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

5013 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,30)(31,32);(1,3,5,4)(2,6,8,7)(11,15)(13,16)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31);(1,3,5,4)(2,7,8,6)(9,11)(10,13)(12,15)(14,16)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31);(10,14)(11,15)(20,29)(23,28)(26,32)(30,31);(11,15)(13,16)(18,24)(22,25)(23,28)(30,31);(1,4,5,3)(2,6,8,7)

Polycyclic presentation:

2^9|b^2=g,c^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dhi,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, 7609468);