#7551021

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
prime radical	C₂×#3280	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2205 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)(15,21)(16,19)(17,25)(18,24)(20,22)(23,28)(26,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,22,27,19,23,18,25)(14,17,24,28,16,26,20,21);(6,9)(10,11)(13,16,19,14)(15,17,23,26)(18,24,22,20)(21,27,28,25);(2,4)(5,7,8,12)(6,10,9,11)(14,16)(15,23)(20,24)(25,27);(13,17,22,28,19,26,18,21)(14,15,24,27,16,23,20,25)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=gi,e^2=fi,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, 7458217);