#7722692

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

4850 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=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, 7629888);