#7671910

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
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5206 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,e^2=i,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, 7579106);