#6483477

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2042 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)(7,12)(8,10)(9,11)(13,14,20,25)(15,26,30,40)(16,31,33,41)(17,23,35,38)(18,24,19,37)(21,28,39,29)(22,27,42,32)(34,43,36,44);(1,2,3,4)(7,8)(10,12)(13,15,18,28,20,30,19,29)(14,21,37,40,25,39,24,26)(16,17,27,36,33,35,32,34)(22,38,41,44,42,23,31,43);(5,7,9,8)(6,10,11,12)(13,16,19,32,20,33,18,27)(14,22,24,31,25,42,37,41)(15,17,29,34,30,35,28,36)(21,38,26,43,39,23,40,44);(5,8,9,7)(6,10,11,12)(13,17)(14,23)(15,27)(16,29)(18,36)(19,34)(20,35)(21,31)(22,40)(24,44)(25,38)(26,42)(28,33)(30,32)(37,43)(39,41);(6,11)(10,12)(13,18,20,19)(14,24,25,37)(15,28,30,29)(16,32,33,27)(17,34,35,36)(21,26,39,40)(22,41,42,31)(23,43,38,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,d^2=h,e^2=i,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=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6390673);