#8701739

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

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

6717 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234863, #72234889
maximal normal subgroup of	#72234863, #72234889

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8608935);