#365765

order	512
nilpotency class	3
isoclinism	isoclinic class of #24985
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1292 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=ef,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,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, 272961);