#230423

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2361
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=eh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,f^a=fi,d^b=de,e^b=eh,g^b=gh,d^c=dh,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137619);