#499246

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

909 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321436, #72321820, #72321832
maximal normal subgroup of	#72321436, #72321820, #72321832
maximal quotient of	#72322603, #72322609

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406442);