#353985

order	512
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1942 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251308, #72251322, #72251323, …
maximal normal subgroup of	#72251308, #72251322, #72251323, …
maximal quotient of	#72314605

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261181);