#7517098

order	512
nilpotency class	3
isoclinism	isoclinic class of #7513980
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	#59022	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1971 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7424294);