#7685391

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	#36263	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5021 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=hi,f^2=gi,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7592587);