#6524750

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500060
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3552	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2954 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)(7,10)(8,12)(9,11)(13,17)(14,16)(15,18)(19,20)(21,22,26,30,27,24,25,29)(23,31,34,35,33,36,32,28);(2,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23)(22,28)(24,35)(25,34)(26,32)(27,33)(29,31)(30,36);(6,11)(7,13)(8,15)(16,20)(21,24,27,22)(23,28,33,35)(25,29,26,30)(31,34,36,32);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,16,17,20)(21,25,27,26)(22,29,24,30)(23,32,33,34)(28,36,35,31);(7,13)(10,17)(14,19)(16,20)(23,33)(28,35)(31,36)(32,34)

Polycyclic presentation:

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

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, 6431946);