#7631707

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	#2403	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5178 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538903);