#185221

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	C₂×#3511	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2354 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)(5,6,9,15,10,14,8,13)(7,12,16,20,17,11,18,19)(22,27)(24,25)(28,34)(29,33)(30,32)(35,36);(5,7)(6,11)(8,16)(9,18)(10,17)(12,14)(13,19)(15,20)(21,22)(23,28)(24,27)(25,33)(26,29)(30,35)(31,36)(32,34);(2,4)(5,7)(6,12)(8,18)(9,16)(10,17)(11,14)(13,19)(15,20)(21,23,26,31)(22,28,29,36)(24,32,25,30)(27,34,33,35);(5,8,10,9)(6,13,14,15)(7,16,17,18)(11,19,12,20)(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=i,d^2=h,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, 92417);