#6459789

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	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3077 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)(7,9)(8,10)(11,12)(13,14,20,26)(15,27,31,38)(16,32,34,42)(17,23,33,40)(18,25,19,24)(21,30,39,29)(22,35,41,28)(36,44,37,43);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,19,30,20,31,18,29)(14,21,25,38,26,39,24,27)(16,33,28,36,34,17,35,37)(22,40,32,43,41,23,42,44);(6,10)(7,11)(13,16,18,35,20,34,19,28)(14,22,24,42,26,41,25,32)(15,17,29,36,31,33,30,37)(21,23,27,43,39,40,38,44);(7,11)(9,12)(13,17)(14,23)(15,28)(16,29)(18,36)(19,37)(20,33)(21,32)(22,27)(24,43)(25,44)(26,40)(30,34)(31,35)(38,41)(39,42);(16,34)(17,33)(22,41)(23,40)(28,35)(32,42)(36,37)(43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gh,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, 6366985);