#497111

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	#2397	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

704 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=ce,d^c=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, 404307);