#7517444

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#35752	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1894 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)(7,12)(8,10)(9,11)(13,14,18,17)(15,22,25,20)(16,21,27,19)(23,28,24,26);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,18,25)(14,20,17,22)(16,23,27,24)(19,28,21,26);(2,4)(13,16)(14,21)(15,23)(17,19)(18,27)(20,26)(22,28)(24,25);(2,4)(5,8,9,7)(6,10,11,12)(13,17,18,14)(15,24,25,23)(16,21,27,19)(20,28,22,26);(6,11)(10,12)(14,17)(16,27)(20,22)(23,24)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7424640);