#7649131

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

5876 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

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, 7556327);