#387434

order	512
nilpotency class	3
isoclinism	isoclinic class of #31076
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2684
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1180 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294630);