#6374282

order	512
nilpotency class	3
isoclinism	isoclinic class of #60753
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄²

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	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4146 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)(7,11)(8,10)(9,12)(13,14)(15,19)(16,26)(17,29)(18,21)(20,27)(22,31)(23,37)(24,33)(25,41)(28,36)(30,35)(32,39)(34,40)(38,44)(42,43);(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,23)(17,25)(18,24)(20,32)(21,33)(22,34)(26,37)(27,39)(28,38)(29,41)(30,42)(31,40)(35,43)(36,44);(7,12)(9,11)(13,16,18,28)(14,20,21,35)(15,23,24,38)(17,30,31,27)(19,32,33,43)(22,36,29,26)(25,42,40,39)(34,44,41,37);(6,10)(9,11)(15,24)(19,33)(23,38)(25,40)(32,43)(34,41)(37,44)(39,42);(14,21)(19,33)(20,35)(22,29)(26,36)(32,43)(34,41)(37,44)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^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, 6281478);