#7700059

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

5453 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7607255);