#7551004

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2174 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:

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

Polycyclic presentation:

2^9|b^2=fi,d^2=gi,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7458200);