#7631944

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	#35761	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5487 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)(6,7)(10,13)(14,16)(17,18)(19,20)(21,25)(22,26)(23,24)(27,28)(29,31)(30,32);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,24)(18,23,26,20)(21,29,30,27)(25,28,32,31);(1,3,5,2)(4,7,8,6)(17,20)(18,24)(19,26)(21,28)(22,23)(25,29)(27,32)(30,31);(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31);(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,29)(30,32);(1,4,5,8)(2,6,3,7)(9,11,12,15)(10,14,13,16)(17,21,22,30)(18,25,26,32)(19,27,24,29)(20,28,23,31)

Polycyclic presentation:

2^9|b^2=h,c^2=hi,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, 7539140);