#6672135

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3119	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1802 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2)(3,4)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(24,32)(29,36)(31,34)(33,35);(5,7,9,15)(6,10,12,13)(8,14,17,20)(11,18,16,19)(21,22,26,30)(23,28,25,27)(24,33,32,35)(29,34,36,31);(2,4)(22,27)(23,25)(28,30)(29,33)(31,34)(35,36);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,23,26,25)(22,28,30,27)(24,31,32,34)(29,35,36,33);(7,15)(10,13)(14,20)(18,19)(21,24,26,32)(22,29,30,36)(23,31,25,34)(27,33,28,35)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257048, #72257108, #72257173
maximal normal subgroup of	#72257048, #72257108, #72257173
maximal quotient of	#72264490

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6579331);