#314613

order	512
nilpotency class	3
isoclinism	isoclinic class of #251800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2482
prime radical	#3449	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

637 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 637 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=e,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,f^b=fh,e^c=eh,f^c=fhi,g^c=gh,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307188, #72307191, #72307239
maximal normal subgroup of	#72307188, #72307191, #72307239

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 221809);