#7515545

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	#23962	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1892 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)(7,12)(8,10)(9,11)(13,14)(15,16)(17,24)(18,23)(19,21)(20,26)(22,27)(25,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,18,26)(14,20,23,16)(17,25,19,22)(21,28,24,27);(5,8,9,7)(6,10,11,12)(14,21)(15,22)(17,19)(23,24)(25,26)(27,28);(2,4)(13,16)(14,22)(15,21)(17,27)(18,20)(19,28)(23,25)(24,26);(6,11)(10,12)(13,17,18,19)(14,21,23,24)(15,25,26,22)(16,27,20,28)

Polycyclic presentation:

2^9|b^2=f,c^2=fi,e^2=i,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, 7422741);