#7709809

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	#39189	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4783 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7617005);