#6399452

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3277	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2188 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,g^c=gi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6306648);