#333380

order	512
nilpotency class	3
isoclinism	isoclinic class of #333271
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#555	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

729 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240576);