#358596

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1670 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)(15,25,27,36)(16,30,34,40)(17,23)(18,24)(19,22)(20,28,37,29)(21,33,39,32)(26,41)(31,44)(35,42)(38,43);(2,4)(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,31)(17,29)(18,28)(19,27)(21,38)(22,37)(23,36)(24,25)(26,33)(30,35)(32,42)(34,43)(39,44)(40,41);(2,4)(5,8,9,7)(6,10,11,12)(13,16)(14,21)(15,26)(17,33)(18,32)(19,34)(20,35)(22,39)(23,30)(24,40)(25,38)(27,42)(28,31)(29,43)(36,44)(37,41);(14,22)(15,27)(21,39)(23,24)(26,42)(28,29)(30,40)(31,43)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265792);