#364807

order	512
nilpotency class	3
isoclinism	isoclinic class of #23876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1348 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=de,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, 272003);