#353224

order	512
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₄	C₂×C₄³
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1642 conjugacy classes of subgroups; only the 882 normal subgroups are shown.

normal subgroups only

This group has 1642 conjugacy classes of subgroups — too many to draw. Only its 882 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=ci,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260420);