#6401013

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1762 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=fi,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,g^c=gi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6308209);