#277221

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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

649 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)(5,6,8,12,9,11,17,19)(7,13,14,18,15,20,16,10)(21,22,24,29,25,30,26,28)(23,31,33,36,34,27,32,35);(1,2,4,3)(21,23)(22,27)(24,33)(25,34)(26,32)(28,36)(29,35)(30,31);(5,7,9,15)(6,10,11,18)(8,16,17,14)(12,20,19,13)(22,28)(23,32)(24,26)(27,31)(29,30)(33,34);(1,3,4,2)(6,11)(10,18)(12,19)(13,20)(21,24,25,26)(22,29,30,28)(23,33,34,32)(27,35,31,36)

Polycyclic presentation:

2^9|a^2=f,b^2=ei,c^2=hi,d^2=e,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, 184417);