#185189

order	512
nilpotency class	3
isoclinism	isoclinic class of #160780
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×#3444	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2552 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)(5,6,9,13,10,14,8,12)(7,15,16,20,17,11,18,19)(21,22)(23,30)(24,28)(25,33)(26,29)(27,31)(32,35)(34,36);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23)(22,27)(24,31)(25,34)(26,32)(28,35)(29,36)(30,33);(2,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,24,26,25)(22,28,29,33)(23,31,32,34)(27,35,36,30);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(22,29)(27,36)(28,33)(30,35)

Polycyclic presentation:

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

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, 92385);