#7565190

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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	#44876	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|c^2=fgi,d^2=fg,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7472386);