#7710016

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	#59040	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4754 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7617212);