#7517586

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	#44869	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1939 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,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, 7424782);