#7718251

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	#51322	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4582 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:

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

Polycyclic presentation:

2^9|d^2=gh,e^2=hi,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,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, 7625447);