#7726838

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	#45079	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4728 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=h,e^2=hi,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, 7634034);