#187144

order	512
nilpotency class	3
isoclinism	isoclinic class of #187021
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	#62824	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2193 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235011, #72235012, #72235014, …
maximal normal subgroup of	#72235011, #72235012, #72235014, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94340);