#367479

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1160 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)(5,6,10,14,11,15,9,23)(7,12,17,27,18,28,24,26)(8,19,22,30,21,34,20,13)(16,31,32,25,33,36,35,29);(2,4)(5,7)(6,12)(8,16,21,33)(9,24)(10,17)(11,18)(13,29,30,25)(14,27)(15,28)(19,31,34,36)(20,35,22,32)(23,26);(5,8,9,20,11,21,10,22)(6,13,14,19,15,30,23,34)(7,16,17,32,18,33,24,35)(12,25,26,31,28,29,27,36);(2,4)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274675);