#245872

order	512
nilpotency class	3
isoclinism	isoclinic class of #245807
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

1017 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=i,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,d^b=dh,e^b=eh,e^c=ei,f^c=fi,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, 153068);