#7516207

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	#45103	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2109 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)(7,10)(8,9)(11,12)(13,14,17,21)(15,22,25,19)(16,26,18,20)(23,24,28,27);(1,2)(3,4)(6,9)(10,12)(13,14,17,21)(15,23,25,28)(16,20,18,26)(19,27,22,24);(2,4)(13,14)(15,19)(16,26)(17,21)(18,20)(22,25)(23,24)(27,28);(2,4)(5,7,8,11)(6,10,9,12)(13,15,17,25)(14,19,21,22)(16,24,18,27)(20,28,26,23);(6,9)(10,12)(13,16,17,18)(14,20,21,26)(15,24,25,27)(19,28,22,23)

Polycyclic presentation:

2^9|a^2=i,b^2=i,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,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, 7423403);