#263401

order	512
nilpotency class	3
isoclinism	isoclinic class of #263395
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62365	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

818 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:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=i,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fhi,d^c=dhi,f^c=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306295, #72306303, #72306316
maximal normal subgroup of	#72306295, #72306303, #72306316

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170597);