#6379697

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

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	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2822 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6286893);