#6477694

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470349
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2086 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314727, #72314731, #72314769, …
maximal normal subgroup of	#72314727, #72314731, #72314769, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6384890);