#8105570

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5785 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304020
maximal normal subgroup of	#72304020
maximal quotient of	#72303956

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8012766);