#6467519

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25777	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2300 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374715);