#382380

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂×#3449	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3670 conjugacy classes of subgroups; only the 362 normal subgroups are shown.

normal subgroups only

This group has 3670 conjugacy classes of subgroups — too many to draw. Only its 362 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) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,27)(23,28)(24,26);(6,9)(10,12)(13,15)(14,18)(16,22)(17,24)(19,27)(20,26)(21,28)(23,25);(2,4)(5,7)(6,10)(8,11)(9,12)(14,19,20,21)(15,22,24,23);(7,11)(10,12)(14,20)(18,26)(19,21)(27,28)

Polycyclic presentation:

2^9|c^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,d^c=de

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289576);