#298129

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
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

1020 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)(8,15)(10,16)(12,18)(14,19)(17,20)(21,22,25,27)(23,31,33,36)(24,30,26,29)(28,32,35,34);(1,2,4,3)(5,6,9,11)(7,10,14,17)(8,12,15,18)(13,16,19,20)(22,27)(23,32)(24,26)(28,31)(33,34)(35,36);(5,7,8,13,9,14,15,19)(6,10,12,16,11,17,18,20)(21,23,24,34,25,33,26,32)(22,28,29,31,27,35,30,36);(1,3,4,2)(6,11)(10,17)(12,18)(16,20)(21,24,25,26)(22,29,27,30)(23,32,33,34)(28,36,35,31)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=f,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=di

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205325);