#6487639

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×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×#3486	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4042 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)(8,11)(9,12)(13,14,20,25)(15,26,16,30)(17,22,34,39)(18,23,19,24)(21,28,36,29)(27,40,31,43)(32,37,33,38)(35,42,44,41);(2,4)(7,8)(10,11)(13,15,20,16)(14,21,25,36)(17,27,34,31)(18,28,19,29)(22,35,39,44)(23,26,24,30)(32,41,33,42)(37,43,38,40);(5,7,9,8)(6,10,12,11)(13,16,20,15)(14,21,25,36)(17,31,34,27)(18,29,19,28)(22,35,39,44)(23,26,24,30)(32,42,33,41)(37,43,38,40);(5,8,9,7)(6,11,12,10)(13,17)(14,22)(15,27)(16,31)(18,33)(19,32)(20,34)(21,35)(23,38)(24,37)(25,39)(26,40)(28,42)(29,41)(30,43)(36,44);(6,12)(10,11)(13,18,20,19)(14,23,25,24)(15,28,16,29)(17,32,34,33)(21,26,36,30)(22,37,39,38)(27,41,31,42)(35,43,44,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,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,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6394835);