#7630819

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5024 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=h,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, 7538015);