#184838

order	512
nilpotency class	3
isoclinism	isoclinic class of #184831
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2188 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 52 points:

(1,2)(3,4)(5,6,8,11,9,12,15,18)(7,10,14,17,13,16,19,20)(21,22)(23,31,34,27)(24,28)(25,29)(26,30)(32,45,48,39)(33,46,50,40)(35,41)(36,42)(37,43)(38,44)(47,52,49,51);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,34)(22,27,30,31)(24,32,36,48)(25,33,38,50)(28,39,42,45)(29,40,44,46)(35,47,37,49)(41,51,43,52);(2,4)(21,24,26,36)(22,28,30,42)(23,32,34,48)(25,37,38,35)(27,39,31,45)(29,43,44,41)(33,49,50,47)(40,52,46,51);(7,13)(10,16)(14,19)(17,20)(21,25,26,38)(22,29,30,44)(23,33,34,50)(24,35,36,37)(27,40,31,46)(28,41,42,43)(32,47,48,49)(39,51,45,52)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=di,g^b=gh,d^c=dh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92034);