#7517535

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7424731);