#7675648

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

5964 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7582844);