#354712

order	512
nilpotency class	3
isoclinism	isoclinic class of #3068
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1030 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250362, #72250370, #72250386, …
maximal normal subgroup of	#72250362, #72250370, #72250386, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261908);