#7641745

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

4996 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)(19,23)(20,28)(21,26)(22,25)(24,29)(27,30)(31,32);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19,22,26)(18,23,25,21)(20,27,29,32)(24,31,28,30);(10,14)(11,15)(17,20)(18,24)(19,27)(21,30)(22,29)(23,31)(25,28)(26,32);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(18,25)(19,26)(24,28)(27,32);(1,4,5,3)(2,6,8,7)(20,29)(24,28)(27,32)(30,31);(17,21,22,23)(18,26,25,19)(20,30,29,31)(24,32,28,27)

Polycyclic presentation:

2^9|b^2=gh,d^2=g,e^2=g,f^2=hi,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.

Relationships

maximal subgroup of	#72303576
maximal normal subgroup of	#72303576
maximal quotient of	#72303666

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7548941);