#184825

order	512
nilpotency class	3
isoclinism	isoclinic class of #184811
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=di,g^b=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92021);