#7671223

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

5244 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=gh,e^2=h,f^2=h,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, 7578419);