#7520435

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#35772	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1894 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)(13,14,18,15)(16,23,25,24)(17,20,28,21)(19,27,22,26);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,18,14)(16,24,25,23)(17,21,28,20)(19,26,22,27);(5,7,9,8)(6,11,12,10)(13,16)(14,19)(15,22)(17,27)(18,25)(20,23)(21,24)(26,28);(2,4)(14,20)(15,21)(17,28)(19,23)(22,24)(26,27);(5,8,9,7)(6,11,12,10)(16,25)(19,22)(23,24)(26,27)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,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, 7427631);