#7561672

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514000
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	#59810	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2243 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:

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

Polycyclic presentation:

2^9|c^2=fg,d^2=fg,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^d=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, 7468868);