#8509009

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

6123 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=hi,d^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234887, #72234891, #72234893
maximal normal subgroup of	#72234887, #72234891, #72234893
maximal quotient of	#72262131

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416205);