#334457

order	512
nilpotency class	3
isoclinism	isoclinic class of #331791
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×#547	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

767 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=h,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=dei,e^b=eh,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 241653);