#7666583

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5007 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7573779);