#7773587

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

5592 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7680783);