#6375859

order	512
nilpotency class	3
isoclinism	isoclinic class of #60753
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

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×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2164 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)(6,9)(7,11)(13,14)(15,25)(16,30)(17,24)(18,22)(19,23)(20,27)(21,32)(26,41)(28,36)(29,37)(31,44)(33,40)(34,39)(35,43)(38,42);(1,3,4,2)(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,31)(17,28)(18,27)(19,29)(21,38)(22,25)(23,36)(24,37)(26,33)(30,41)(32,43)(34,42)(35,39)(40,44);(6,9)(10,12)(13,16)(14,21)(15,26)(17,32)(18,33)(19,34)(20,35)(22,39)(23,30)(24,40)(25,38)(27,31)(28,42)(29,43)(36,44)(37,41);(5,7,8,11)(6,10,9,12)(14,22)(15,27)(21,39)(23,24)(26,31)(28,29)(30,40)(42,43);(13,17,18,19)(14,23,22,24)(15,28,27,29)(16,32,33,34)(20,36,25,37)(21,30,39,40)(26,42,31,43)(35,44,38,41)

Polycyclic presentation:

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

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, 6283055);