#8934366

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

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

8854 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72237717, #72240899, #72240900, …
maximal subgroup of	#72210439, #72210483, #72210487, …
maximal normal subgroup of	#72210439, #72210483, #72210487, …
maximal quotient of	#72230894, #72231761, #72232388, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841562);