#184712

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂⁴
commutator, socle, prime radical	C₂³	C₂²×C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

590 conjugacy classes of subgroups; only the 310 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 310 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) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=e,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=de,c^b=ceh,d^b=dhi,d^c=dh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91908);