#7864677

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	#43906	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5042 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7771873);