#6343619

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343406
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62815	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2884 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,e^2=hi,f^2=i,b^a=bf,e^a=ei,f^a=fh,f^b=fi,g^b=ghi,d^c=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317810, #72318533, #72318685
maximal normal subgroup of	#72317810, #72318533, #72318685

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250815);