#227603

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	#2358
prime radical	#3481	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

629 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287666, #72322026, #72322077
maximal normal subgroup of	#72287666, #72322026, #72322077

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 134799);