#298213

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	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1025 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)(7,13)(8,14)(10,16)(12,17)(15,19)(18,20)(21,22,25,27)(23,31,34,36)(24,30,26,29)(28,32,35,33);(1,2,4,3)(5,6,9,11)(7,10,15,18)(8,12,14,17)(13,16,19,20)(22,27)(23,32)(24,26)(28,31)(33,34)(35,36);(5,7,14,19,9,15,8,13)(6,10,17,20,11,18,12,16)(21,23,26,32,25,34,24,33)(22,28,30,36,27,35,29,31);(1,3,4,2)(6,11)(7,15)(12,17)(13,19)(21,24,25,26)(22,29,27,30)(23,33,34,32)(28,31,35,36)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=fh,d^2=e,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, 205409);