#300634

order	512
nilpotency class	3
isoclinism	isoclinic class of #264043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

987 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207830);