#382525

order	512
nilpotency class	3
isoclinism	isoclinic class of #184805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

2490 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 2490 conjugacy classes of subgroups — too many to draw. Only its 322 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) = 64

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,h^b=hi,d^c=de,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, 289721);