#245803

order	512
nilpotency class	3
isoclinism	isoclinic class of #245711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
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

1028 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152999);