#6454376

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2004 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361572);