#298225

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×#1621	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1119 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)(23,28)(24,33)(25,29)(26,34)(27,30)(31,36)(32,35);(1,2,4,3)(5,6,9,11)(7,10,14,17)(8,12,15,18)(13,16,19,20)(21,23,27,32)(22,28,30,35)(24,25,31,26)(29,36,34,33);(5,7,8,13,9,14,15,19)(6,10,12,16,11,17,18,20)(21,24,26,23,27,31,25,32)(22,28,29,33,30,35,34,36);(1,3,4,2)(6,11)(7,14)(12,18)(13,19)(21,25,27,26)(22,29,30,34)(23,24,32,31)(28,33,35,36)

Polycyclic presentation:

2^9|b^2=eh,c^2=f,d^2=ei,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, 205421);