#7565490

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#25087	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1854 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)(8,12)(9,10)(13,14,19,22)(15,17,21,18)(16,23,26,25)(20,28,24,27);(1,2)(3,4)(6,10)(11,12)(13,15)(14,17)(16,24)(18,22)(19,21)(20,26)(23,27)(25,28);(5,7,9,8)(6,11,10,12)(13,16,17,27,19,26,18,28)(14,20,15,23,22,24,21,25);(2,4)(5,7,9,8)(6,11,10,12)(13,15,17,22,19,21,18,14)(16,25,27,20,26,23,28,24);(5,8,9,7)(6,11,10,12)(16,26)(20,24)(23,25)(27,28)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=fg,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,d^c=di,e^c=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, 7472686);