#7875551

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

5195 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7782747);