#6473349

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470901
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	C₂×#3204	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2332 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)(13,14,19,24)(15,25,29,41)(16,30,32,38)(17,22,34,37)(18,36,26,23)(20,31,39,28)(21,33,40,27)(35,43,44,42);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,26,31,19,29,18,28)(14,20,36,41,24,39,23,25)(16,17,33,44,32,34,27,35)(21,37,38,42,40,22,30,43);(7,11)(9,12)(13,16,18,27,19,32,26,33)(14,21,23,30,24,40,36,38)(15,17,28,35,29,34,31,44)(20,37,25,43,39,22,41,42);(7,11)(9,12)(13,17)(14,22)(15,27)(16,31)(18,35)(19,34)(20,38)(21,25)(23,42)(24,37)(26,44)(28,32)(29,33)(30,39)(36,43)(40,41);(6,10)(9,12)(16,32)(17,34)(21,40)(22,37)(27,33)(30,38)(35,44)(42,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,d^c=di,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, 6380545);