#6345430

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	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#3013	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2965 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1541 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,e^a=ei,f^a=fh,f^b=fh,d^c=di,e^d=eh

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, 6252626);