#7148221

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

1760 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(7,11)(10,12)(13,14,17,21,18,22,31,37)(15,23,25,33,26,36,30,19)(16,27,29,43,28,20,34,35)(24,39,41,32,40,38,44,42);(1,2)(3,4)(6,9)(7,11);(5,6)(7,10)(8,9)(11,12)(13,15,18,26)(14,19,22,33)(16,24,28,40)(17,30,31,25)(20,32,27,42)(21,36,37,23)(29,44,34,41)(35,39,43,38);(2,4)(6,9)(10,12)(13,16,18,28)(14,20,22,27)(15,24,26,40)(17,29,31,34)(19,32,33,42)(21,35,37,43)(23,38,36,39)(25,41,30,44);(5,7,8,11)(6,10,9,12)(16,28)(20,27)(24,40)(29,34)(32,42)(35,43)(38,39)(41,44)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7055417);