#7682194

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

5314 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)(4,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,20)(19,23,24,22);(2,6)(3,7)(17,19,21,24)(18,22,20,23);(1,2)(3,4)(5,6)(7,8)(10,13)(14,16);(2,6)(3,7)(9,10)(11,14)(12,13)(15,16);(1,3,5,7)(2,4,6,8)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,23,24,22);(1,4)(2,7)(3,6)(5,8)

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7589390);