#183125

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1176 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,g^2=i,b^a=be,d^a=di,e^a=eh,g^a=gh,c^b=ch,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, 90321);