#7630855

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	#35737	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4896 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^d=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, 7538051);