#374954

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

930 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=h,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282150);