#7671810

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5600 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7579006);