#7148239

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	#42650	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1784 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)(13,14,19,21,18,24,17,23)(15,20,31,33,27,36,26,35)(16,28,32,38,29,22,30,37)(25,39,42,44,40,34,41,43);(1,2)(3,4)(6,9)(7,11)(13,15,18,27)(14,20,24,36)(16,25,29,40)(17,26,19,31)(21,33,23,35)(22,34,28,39)(30,41,32,42)(37,43,38,44);(5,6)(7,10)(8,9)(11,12)(14,21)(17,19)(20,33)(22,37)(23,24)(26,31)(28,38)(30,32)(34,43)(35,36)(39,44)(41,42);(2,4)(6,9)(10,12)(13,16)(14,22)(15,25)(17,30)(18,29)(19,32)(20,34)(21,37)(23,38)(24,28)(26,41)(27,40)(31,42)(33,43)(35,44)(36,39);(5,7,8,11)(6,10,9,12)(16,29)(22,28)(25,40)(30,32)(34,39)(37,38)(41,42)(43,44)

Polycyclic presentation:

2^9|a^2=gi,b^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, 7055435);