#6436758

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433040
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

2020 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)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,22,25,28)(23,29,31,36)(24,33,30,27)(26,34,35,32);(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23,30,34,25,31,24,32)(22,26,33,36,28,35,27,29);(2,4)(7,13)(10,16)(14,19)(17,20);(6,11)(10,16)(12,18)(17,20)(23,31)(26,35)(29,36)(32,34);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)(23,31)(26,35)(29,36)(32,34)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6343954);