#300710

order	512
nilpotency class	3
isoclinism	isoclinic class of #264123
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	C₂×#550	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1022 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)(5,6,10,14)(7,15,8,18)(9,20,16,13)(11,19,12,17)(21,22)(23,27)(24,32)(25,34)(26,29)(28,33)(30,36)(31,35);(5,7,9,17,10,8,16,19)(6,11,13,15,14,12,20,18)(21,23,26,31)(22,27,29,35)(24,33,30,25)(28,36,34,32);(1,2,3,4)(7,8)(11,12)(15,18)(17,19)(21,24,25,23,26,30,33,31)(22,27,28,32,29,35,34,36);(5,8,9,19,10,7,16,17)(6,12,13,18,14,11,20,15)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=f,d^2=e,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,d^c=dh

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, 207906);