#497115

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

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

664 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 226 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)(13,14,18,24,19,25,31,39,17,23,20,32,16,22,30,38)(15,21,28,36,26,34,40,43,27,35,33,42,29,37,41,44);(5,7,10,6)(8,11,9,12)(13,14,20,32,19,25,30,38,17,23,18,24,16,22,31,39)(15,21,33,42,26,34,41,44,27,35,28,36,29,37,40,43);(1,2,3,4)(5,8,10,9)(6,11,7,12)(13,15,17,27)(14,21,23,35)(16,29,19,26)(18,28,20,33)(22,37,25,34)(24,36,32,42)(30,41,31,40)(38,44,39,43);(5,9,10,8)(6,11,7,12)(13,16,17,19)(14,22,23,25)(15,26,27,29)(18,30,20,31)(21,34,35,37)(24,38,32,39)(28,40,33,41)(36,43,42,44)

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=dei,c^b=cei,d^c=di

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, 404311);