#365025

order	512
nilpotency class	3
isoclinism	isoclinic class of #25440
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	#60450	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1662 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)(5,6,12,18)(7,19,22,15)(8,23)(9,13,21,29)(10,17,11,16)(14,25)(20,34)(24,33)(26,32)(27,31)(28,36)(30,35);(5,7,11,21,12,22,10,9)(6,13,17,15,18,29,16,19)(8,24,25,36,27,35,26,20)(14,30,23,28,32,33,31,34);(2,4)(5,8)(6,14)(7,20)(9,24)(10,25)(11,26)(12,27)(13,28)(15,30)(16,23)(17,31)(18,32)(19,33)(21,35)(22,36)(29,34);(5,9,10,22,12,21,11,7)(6,15,16,13,18,19,17,29)(8,24,25,36,27,35,26,20)(14,30,23,28,32,33,31,34)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272221);