#277273

order	512
nilpotency class	3
isoclinism	isoclinic class of #267291
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
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

680 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)(5,6,9,12,10,13,8,17)(7,14,15,20,16,19,18,11)(21,22,24,28,25,29,26,30)(23,31,32,36,33,27,34,35);(1,2,4,3)(21,23)(22,27)(24,32)(25,33)(26,34)(28,35)(29,31)(30,36);(5,7,10,16)(6,11,13,20)(8,15,9,18)(12,19,17,14)(22,28)(23,32)(24,26)(27,31)(29,30)(33,34);(1,3,4,2)(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14)(21,24,25,26)(22,28,29,30)(23,32,33,34)(27,35,31,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=ei,c^2=hi,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 184469);