#262204

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
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	C₂×H(3,𝔽₄)	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

950 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=h,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 169400);