#382393

order	512
nilpotency class	3
isoclinism	isoclinic class of #184805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#537	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3822 conjugacy classes of subgroups; only the 362 normal subgroups are shown.

normal subgroups only

This group has 3822 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)(15,21)(16,24)(17,20)(18,23)(19,25)(22,26)(27,28);(6,9)(10,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,25)(26,28);(2,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,18,20,21)(16,22)(17,23)(19,26,24,27)(25,28);(7,11)(10,12)(13,16,17,25)(14,19,20,24)(15,22,23,28)(18,26,21,27)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289589);