#7631317

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	#34981	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6246 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 24 points:

(9,10,13,16)(11,12,15,14)(17,18,20,22)(19,21,23,24);(2,5)(6,8)(9,11,13,15)(10,14,16,12)(17,18)(19,21)(20,22)(23,24);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16);(3,7)(6,8)(9,12,13,14)(10,15,16,11)(17,19,20,23)(18,21,22,24);(1,3)(2,6)(4,7)(5,8);(1,2)(3,6)(4,5)(7,8)

Polycyclic presentation:

2^9|a^2=g,b^2=h,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, 7538513);