#7557482

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1789 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7464678);