#399108

order	512
nilpotency class	3
isoclinism	isoclinic class of #393900
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1357 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=efi,c^2=ef,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 306304);