#7630542

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	#35223	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5126 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:

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

Polycyclic presentation:

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

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, 7537738);