#6457368

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2304 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)(5,6,8,10)(7,9,11,12)(13,14)(15,26)(16,31)(17,23)(18,24)(19,37)(20,25)(21,29)(22,27)(28,40)(30,44)(32,39)(33,42)(34,38)(35,41)(36,43);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,19,29,20,30,18,28)(14,21,24,26,25,40,37,44)(16,17,27,33,32,34,35,36)(22,38,31,42,41,23,39,43);(7,11)(9,12)(13,16,18,35,20,32,19,27)(14,22,37,39,25,41,24,31)(15,17,28,36,30,34,29,33)(21,38,44,43,40,23,26,42);(7,11)(9,12)(13,17)(14,23)(15,27)(16,28)(18,36)(19,33)(20,34)(21,39)(22,26)(24,43)(25,38)(29,32)(30,35)(31,40)(37,42)(41,44);(13,18,20,19)(14,24,25,37)(15,28,30,29)(16,27,32,35)(17,33,34,36)(21,26,40,44)(22,39,41,31)(23,42,38,43)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=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, 6364564);