#7604343

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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	#45271	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1846 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)(5,6,9,8)(7,10,12,11)(13,14,20,24)(15,16,22,27)(17,23,19,28)(18,26,25,21);(2,3)(5,7)(6,10)(8,11)(9,12)(13,15,19,25,20,22,17,18)(14,21,23,16,24,26,28,27);(1,2,4,3)(5,8,9,6)(7,11,12,10)(13,16)(14,22)(15,24)(17,21)(18,23)(19,26)(20,27)(25,28);(1,3,4,2)(7,12)(10,11)(13,17,20,19)(14,23,24,28)(15,18,22,25)(16,26,27,21);(13,18,19,15,20,25,17,22)(14,16,23,26,24,27,28,21)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7511539);