#560016

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

1798 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=fg,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 467212);