#7763779

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4999 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7670975);