#185310

order	512
nilpotency class	3
isoclinism	isoclinic class of #185261
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3493	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2216 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18,21,26,22,25,34,39)(19,27,30,40,29,23,33,36)(20,24,32,38,31,37,44,47)(28,35,43,46,42,41,45,48);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19)(18,23)(20,28)(21,33)(22,29)(24,35)(25,27)(26,40)(30,34)(31,42)(32,45)(36,39)(37,41)(38,48)(43,44)(46,47);(17,20,22,31)(18,24,25,37)(19,28,29,42)(21,32,34,44)(23,35,27,41)(26,38,39,47)(30,43,33,45)(36,46,40,48);(2,7)(6,14)(8,15)(9,16)(18,25)(23,27)(24,37)(26,39)(35,41)(36,40)(38,47)(46,48)

Polycyclic presentation:

2^9|a^2=g,c^2=h,e^2=i,g^2=h,b^a=be,c^a=cf,d^a=dhi,e^a=eh,e^b=ei,f^b=fh,g^b=ghi,e^c=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92506);