#8910147

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

6184 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72228020, #72228023
Fitting subgroup of	#72228020, #72228022, #72228024
maximal subgroup of	#72228003, #72228007, #72228008, …
maximal normal subgroup of	#72219285, #72228003, #72228007, …
maximal quotient of	#72219278, #72268012, #72268378, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8817343);