#369239

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1340 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:

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

Polycyclic presentation:

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

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, 276435);