#7533711

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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

2119 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7440907);