#185086

order	512
nilpotency class	3
isoclinism	isoclinic class of #160892
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3506	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2566 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)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18)(21,22)(23,30)(24,28)(25,33)(26,29)(27,31)(32,35)(34,36);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23)(22,27)(24,31)(25,34)(26,32)(28,35)(29,36)(30,33);(2,4)(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(21,24,26,25)(22,28,29,33)(23,31,32,34)(27,35,36,30);(8,16)(11,18)(14,20)(17,19)(21,24,26,25)(22,28,29,33)(23,31,32,34)(27,35,36,30)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=eh,e^b=ei,g^b=ghi,d^c=dhi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92282);