#6388351

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60557	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1657 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18,22,27,23,20,32,26)(19,24,30,39,31,28,43,38)(21,25,36,42,35,33,44,41)(29,37,45,48,34,40,46,47);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,16)(10,11)(17,19,22,30,23,31,32,43)(18,24,27,39,20,28,26,38)(21,34,36,46,35,29,44,45)(25,40,42,47,33,37,41,48);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20)(18,23)(19,28)(21,33)(22,26)(24,31)(25,35)(27,32)(29,40)(30,38)(34,37)(36,41)(39,43)(42,44)(45,47)(46,48);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(17,21)(18,25)(19,29)(20,33)(22,36)(23,35)(24,37)(26,41)(27,42)(28,40)(30,45)(31,34)(32,44)(38,47)(39,48)(43,46);(2,10)(7,11)(8,9)(15,16)(21,35)(25,33)(29,34)(36,44)(37,40)(41,42)(45,46)(47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6295547);