#493086

order	512
nilpotency class	4
isoclinism	isoclinic class of #33469
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1128 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=i,e^2=h,f^2=i,g^2=h,h^2=i,b^a=be,e^a=eh,g^a=gh,h^a=hi,c^b=ch,f^b=fi,d^c=di,e^c=ei,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319789, #72319799, C₄×#33471, …
maximal normal subgroup of	#72319789, #72319799, C₄×#33471, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400282);