#503119

order	512
nilpotency class	4
isoclinism	isoclinic class of #501058
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60639	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1289 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1289 conjugacy classes of subgroups — too many to draw. Only its 194 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 — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224877
maximal normal subgroup of	#72224877

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410315);