#7517397

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	#45105	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1986 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)(15,22)(16,20)(17,19)(18,27)(21,25)(23,28)(24,26);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,19,25)(14,21,17,22)(16,26,27,23)(18,24,20,28);(2,4)(13,16)(14,18)(15,23)(17,20)(19,27)(21,28)(22,24)(25,26);(2,4)(5,8,9,7)(6,10,11,12)(13,17,19,14)(15,23,25,26)(16,18,27,20)(21,28,22,24);(6,11)(10,12)(13,18,19,20)(14,16,17,27)(15,24,25,28)(21,26,22,23)

Polycyclic presentation:

2^9|b^2=f,d^2=f,e^2=i,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,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, 7424593);