#332972

order	512
nilpotency class	3
isoclinism	isoclinic class of #332911
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

622 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)(3,10)(4,12)(5,7)(6,15)(8,11)(9,14)(13,16)(17,18,21,27,22,28,23,24)(19,29,30,38,32,25,37,41)(20,33,34,26,35,39,36,42)(31,45,46,43,47,48,44,40);(2,7)(8,16)(9,15)(10,12)(18,24)(19,30)(20,34)(21,23)(26,42)(27,28)(32,37)(35,36)(38,41)(40,45)(43,48)(44,46);(1,3,5,4)(2,8,7,16)(6,13,14,11)(9,12,15,10)(17,19,22,32)(18,25,28,29)(20,31,35,47)(21,30,23,37)(24,38,27,41)(26,40,42,43)(33,48,39,45)(34,46,36,44);(1,4,5,3)(2,8,7,16)(6,11,14,13)(9,12,15,10)(17,20)(18,26)(19,31)(21,36)(22,35)(23,34)(24,39)(25,40)(27,33)(28,42)(29,43)(30,44)(32,47)(37,46)(38,45)(41,48)

Polycyclic presentation:

2^9|a^2=ei,c^2=hi,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=dei,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240168);