#6370593

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×#121	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2074 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:

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

Polycyclic presentation:

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

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, 6277789);