#7672191

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

5102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|c^2=i,d^2=h,e^2=i,f^2=g,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, 7579387);