#7684850

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	#36066	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5476 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=gh,e^2=i,f^2=i,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, 7592046);