#7671233

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

6214 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=h,e^2=h,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, 7578429);