#278788

order	512
nilpotency class	3
isoclinism	isoclinic class of #268763
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

648 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)(5,6,10,12,8,13,9,14)(7,15,18,20,16,11,17,19)(23,29)(26,35)(30,36)(31,34);(1,2,4,3)(5,7,8,16)(6,11,13,15)(9,17,10,18)(12,19,14,20)(21,22,24,27)(23,30,29,36)(25,33,32,28)(26,34,35,31);(6,12)(9,10)(11,19)(13,14)(15,20)(17,18)(21,23,24,29)(22,26,27,35)(25,34,32,31)(28,36,33,30);(1,3,4,2)(6,13)(7,16)(12,14)(17,18)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185984);