#7154764

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#45241	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2071 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)(13,14,19,23)(15,20,18,33)(16,26,29,36)(17,22,32,40)(21,24,37,28)(25,34,31,39)(27,43,30,44)(35,41,38,42);(1,2)(3,4)(6,9)(7,11)(13,15,19,18)(14,20,23,33)(16,24,29,28)(17,25,32,31)(21,26,37,36)(22,34,40,39)(27,41,30,42)(35,43,38,44);(5,6)(7,10)(8,9)(11,12)(13,16,15,24,19,29,18,28)(14,21,33,36,23,37,20,26)(17,30,25,42,32,27,31,41)(22,38,39,43,40,35,34,44);(2,4)(6,9)(10,12)(13,17)(14,22)(15,25)(16,27)(18,31)(19,32)(20,34)(21,35)(23,40)(24,41)(26,43)(28,42)(29,30)(33,39)(36,44)(37,38);(5,7,8,11)(6,10,9,12)(13,15,19,18)(14,20,23,33)(16,28,29,24)(17,31,32,25)(21,36,37,26)(22,39,40,34)(27,41,30,42)(35,43,38,44)

Polycyclic presentation:

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

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, 7061960);