#7653717

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	#36576	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5074 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7560913);