#7520083

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514000
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#30458	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1905 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7427279);