#298232

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
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) = 512

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

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

Polycyclic presentation:

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

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, 205428);