#356994

order	512
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2361
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1406 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314212, #72314246, #72314268
maximal normal subgroup of	#72314212, #72314246, #72314268

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264190);