#187554

order	512
nilpotency class	3
isoclinism	isoclinic class of #187428
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62759	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2190 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)(3,10,12,6)(4,9)(5,8)(7,11,15,13)(14,16)(17,18,22,26,23,28,21,27)(19,29,31,24,33,39,32,38)(20,25,35,40,36,41,34,42)(30,44,45,48,46,43,47,37);(1,3)(2,6)(4,11)(5,12)(7,9)(8,10)(13,14)(15,16)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,37)(26,29)(27,39)(28,38)(34,45)(35,47)(36,46)(40,43)(41,48)(42,44);(1,3)(2,7)(4,13)(5,12)(6,9)(8,15)(10,16)(11,14)(17,20)(18,25)(19,30)(21,34)(22,35)(23,36)(24,37)(26,40)(27,42)(28,41)(29,43)(31,45)(32,47)(33,46)(38,48)(39,44);(2,8)(6,10)(7,15)(9,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,34,36,35)(24,29,38,39)(25,40,41,42)(30,45,46,47)(37,43,48,44)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286596, #72286729, #72286769
maximal normal subgroup of	#72286596, #72286729, #72286769

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94750);