#185081

order	512
nilpotency class	3
isoclinism	isoclinic class of #160892
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2268 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)(7,14)(8,12)(9,17)(10,13)(11,15)(16,19)(18,20)(22,26,28,31)(27,33,34,36);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,20)(14,17)(21,22)(23,27)(24,31)(25,28)(26,32)(29,34)(30,36)(33,35);(2,4)(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14)(21,23)(22,27)(24,30)(25,29)(26,33)(28,34)(31,36)(32,35);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14)(23,29)(27,34)(30,35)(33,36)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,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, 92277);