#7541940

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518952
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	#25422	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1867 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=g,d^2=i,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^c=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, 7449136);