#7527415

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	#44899	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2078 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,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

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, 7434611);