#6487635

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×#3485	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4150 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)(13,14,19,25)(15,26,30,36)(16,22,33,38)(17,23,18,24)(20,28,37,29)(21,32,39,31)(27,41,44,34)(35,43,40,42);(2,4)(7,8)(10,11)(13,15,19,30)(14,20,25,37)(16,27,33,44)(17,28,18,29)(21,34,39,41)(22,35,38,40)(23,26,24,36)(31,42,32,43);(5,7,9,8)(6,10,12,11)(13,16)(14,21)(15,27)(17,31)(18,32)(19,33)(20,34)(22,24)(23,38)(25,39)(26,40)(28,42)(29,43)(30,44)(35,36)(37,41);(5,8,9,7)(6,11,12,10)(13,16)(14,22)(15,27)(17,32)(18,31)(19,33)(20,35)(21,24)(23,39)(25,38)(26,41)(28,43)(29,42)(30,44)(34,36)(37,40);(6,12)(10,11)(13,17,19,18)(14,23,25,24)(15,28,30,29)(16,31,33,32)(20,26,37,36)(21,38,39,22)(27,42,44,43)(34,40,41,35)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,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, 6394831);