#185091

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

2480 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)(21,22)(23,29)(24,32)(25,27)(26,30)(28,34)(31,36)(33,35);(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23,25,31)(22,26,27,35)(24,33,34,30)(28,36,32,29);(2,4)(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20);(6,12)(8,16)(10,13)(14,20)(22,27)(26,35)(28,32)(29,36)

Polycyclic presentation:

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

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, 92287);