#6448936

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2307 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6356132);