#7631953

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	#35000	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5468 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^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, 7539149);