#354483

order	512
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1494 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)(7,17,22,31)(8,9)(10,14)(11,16)(12,20,30,21)(13,24)(15,28)(18,29,34,35)(19,33,32,36)(23,26)(25,27);(1,2,3,4)(5,7)(6,12)(8,18)(9,19)(10,21)(11,22)(13,29)(14,31)(15,17)(16,30)(20,28)(23,36)(24,33)(25,34)(26,35)(27,32);(5,8)(6,13)(7,18)(9,14)(10,23)(11,25)(12,29)(15,27)(16,26)(17,32)(19,31)(20,33)(21,36)(22,34)(24,28)(30,35);(5,9,10,26,11,27,28,13)(6,8,14,23,16,25,15,24)(7,19,21,35,22,32,20,29)(12,18,31,36,30,34,17,33)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214887, #72312195, #72312202, …
maximal normal subgroup of	#72214887, #72312195, #72312202, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261679);