#185073

order	512
nilpotency class	3
isoclinism	isoclinic class of #163899
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2338 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)(22,27,29,33)(28,34,36,35);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,22)(23,28)(24,27)(25,33)(26,29)(30,35)(31,36)(32,34);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,23)(22,28)(24,32)(25,30)(26,31)(27,34)(29,36)(33,35);(21,24,26,25)(22,27,29,33)(23,30,31,32)(28,35,36,34)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92269);