#7561465

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2410 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=fgi,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, 7468661);