#370289

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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×C₄⋊C₄
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

846 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)(5,6,11,16,12,17,10,27)(7,18,21,31,22,30,20,13)(8,23,25,33,26,15,9,14)(19,28,24,34,36,29,35,32);(5,7,12,22)(6,13)(8,24,26,35)(9,19,25,36)(10,20,11,21)(14,32)(15,29)(16,18)(17,31)(23,28)(27,30)(33,34);(2,4)(5,8,10,9,12,26,11,25)(6,14,16,23,17,33,27,15)(7,19,21,24,22,36,20,35)(13,28,30,32,31,29,18,34);(5,9,10,26,12,25,11,8)(6,15,16,14,17,23,27,33)(7,19,20,35,22,36,21,24)(13,29,18,32,31,28,30,34)

Polycyclic presentation:

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

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, 277485);