#6452008

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2754 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:

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

Polycyclic presentation:

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

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, 6359204);