#367933

order	512
nilpotency class	3
isoclinism	isoclinic class of #25440
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	#2361
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1096 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)(5,6,11,16,12,17,10,26)(7,18,20,28,21,15,9,13)(8,14,24,30,25,32,23,31)(19,33,34,36,35,29,22,27);(2,4)(5,7,11,20,12,21,10,9)(6,13,26,15,17,28,16,18)(8,22,23,35,25,34,24,19)(14,29,30,27,32,33,31,36);(2,4)(5,8)(6,14)(7,19)(9,22)(10,23)(11,24)(12,25)(13,27)(15,29)(16,30)(17,32)(18,33)(20,34)(21,35)(26,31)(28,36);(2,4)(5,9,10,21,12,20,11,7)(6,15,16,13,17,18,26,28)(8,22,23,35,25,34,24,19)(14,29,30,27,32,33,31,36)

Polycyclic presentation:

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

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, 275129);