#10366274

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

5634 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262870, #72262922, #72263018
maximal normal subgroup of	#72262870, #72262922, #72263018

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10273470);