#185198

order	512
nilpotency class	3
isoclinism	isoclinic class of #160892
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3508	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2589 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)(5,6,8,12,9,10,15,17)(7,11,14,18,13,16,19,20)(21,22)(23,28)(24,29)(25,34)(26,30)(27,31)(32,35)(33,36);(6,10)(8,15)(11,16)(14,19)(21,23)(22,27)(24,31)(25,33)(26,32)(28,34)(29,35)(30,36);(2,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,23)(22,28)(24,33)(25,31)(26,32)(27,34)(29,36)(30,35);(7,13)(11,16)(14,19)(18,20)(21,24,26,25)(22,29,30,34)(23,31,32,33)(27,35,36,28)

Polycyclic presentation:

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

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, 92394);