#94284

order	512
nilpotency class	5
isoclinism	isoclinic class of #7398
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1343
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂×#408	C₂²
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂×#408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

352 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,e^2=i,f^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,c^b=cg,g^b=gh,h^b=hi,e^c=eh,g^e=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1480);