#6370465

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2362 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=g,g^2=i,b^a=bf,c^a=ci,d^a=df,e^a=ei,c^b=cfi,d^b=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277661);