#347025

order	512
nilpotency class	3
isoclinism	isoclinic class of #347002
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62878	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2146 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)(8,14,16,13)(17,18,23,28)(19,29,33,37)(20,26,34,42)(21,27,22,25)(24,32,39,31)(30,44,45,43)(35,40,36,41)(38,47,48,46);(1,3,7,4)(2,8,11,16)(5,13,6,14)(9,15,12,10)(17,19,23,33)(18,24,28,39)(20,30,34,45)(21,32,22,31)(25,37,27,29)(26,38,42,48)(35,47,36,46)(40,44,41,43);(1,4,7,3)(2,9,11,12)(5,13,6,14)(8,15,16,10)(18,25)(20,34)(21,22)(24,37)(26,41)(27,28)(29,39)(30,45)(31,32)(38,43)(40,42)(44,48);(1,5,7,6)(2,10,11,15)(3,13,4,14)(8,9,16,12)(17,20,23,34)(18,26,28,42)(19,30,33,45)(21,36,22,35)(24,38,39,48)(25,40,27,41)(29,43,37,44)(31,47,32,46)

Polycyclic presentation:

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

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, 254221);