#7551168

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	C₂×#3264	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2244 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)(8,9)(10,12)(13,14,19,23)(15,16,21,25)(17,26,24,20)(18,27,28,22);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,24,19,21,28,17)(14,20,22,16,23,26,27,25);(6,9)(10,11)(13,16)(14,21)(15,23)(17,22)(18,26)(19,25)(20,28)(24,27);(2,4)(5,7,8,12)(6,10,9,11)(15,21)(16,25)(17,24)(20,26);(13,17,18,15,19,24,28,21)(14,16,22,26,23,25,27,20)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7458364);