#7715218

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

5396 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622414);