#262552

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

963 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=h,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 169748);