#185220

order	512
nilpotency class	3
isoclinism	isoclinic class of #161262
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	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2170 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6)(7,13)(8,10)(9,17)(11,15)(12,19)(14,18)(16,20)(21,22,25,30,26,29,24,28)(23,31,32,27,34,35,33,36);(5,7,10,16)(6,11,8,14)(9,18,19,15)(12,20,17,13)(21,23)(22,27)(24,32)(25,33)(26,34)(28,35)(29,36)(30,31);(2,4)(5,8,10,6)(7,14,16,11)(9,17,19,12)(13,15,20,18)(21,24,26,25)(22,28,29,30)(23,32,34,33)(27,35,36,31);(6,8)(11,14)(12,17)(13,20)(22,29)(27,36)(28,30)(31,35)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92416);