#7533661

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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

2097 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)(8,10)(9,12)(14,18)(16,20)(19,24)(21,25)(23,27)(26,28);(1,2)(3,4)(7,12)(9,11)(13,14,16,18,17,21,20,25)(15,19,23,24,22,26,27,28);(5,7,8,12)(6,9,10,11)(13,15,17,22)(14,19,21,26)(16,23,20,27)(18,24,25,28);(2,4)(6,10)(9,11)(15,22)(19,26)(23,27)(24,28);(13,14,20,25,17,21,16,18)(15,19,27,28,22,26,23,24)

Polycyclic presentation:

2^9|b^2=f,c^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,e^b=ei,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7440857);