#247027

order	512
nilpotency class	3
isoclinism	isoclinic class of #247017
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	#51752	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1078 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154223);