#7670692

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
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4932 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577888);