#7682116

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

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

5508 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7589312);