#7520878

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	#36228	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1884 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fi,e^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, 7428074);