#438235

order	512
nilpotency class	3
isoclinism	isoclinic class of #438234
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15616
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

863 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 863 conjugacy classes of subgroups — too many to draw. Only its 208 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219486, #72219494, #72338839
maximal normal subgroup of	#72219486, #72219494, #72338839

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 345431);