#10361117

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

5177 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(9,10,13,15)(11,12,16,14)(17,18)(19,28)(20,26)(21,32)(22,30)(23,27)(24,31)(25,29);(1,2)(3,7)(4,6)(5,8)(9,11,13,16)(10,14,15,12)(17,19)(18,22)(20,21)(23,30)(24,32)(25,26)(27,28)(29,31);(1,2,4,6)(3,5,8,7)(9,11)(10,12)(13,16)(14,15)(17,20)(18,24)(19,29)(21,30)(22,25)(23,31)(26,27)(28,32);(1,2,4,6)(3,5,8,7)(9,12)(10,11)(13,14)(15,16)(17,21,23,29)(18,25,27,32)(19,20,30,31)(22,26,28,24);(1,2)(3,5)(4,6)(7,8)(17,20)(18,26)(19,21)(22,25)(23,31)(24,27)(28,32)(29,30);(1,3)(2,5)(4,8)(6,7)(17,22)(18,19)(20,25)(21,26)(23,28)(24,29)(27,30)(31,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gi,d^2=ghi,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	#72340979
maximal normal subgroup of	#72340979

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10268313);