#354476

order	512
nilpotency class	3
isoclinism	isoclinic class of #3019
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#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×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1480 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)(7,17,21,32)(8,22,24,13)(9,18)(10,15)(11,16)(12,20,30,27)(14,26)(19,23,33,28)(25,35)(29,34)(31,36);(1,2,3,4)(5,7)(6,12)(8,18)(9,19)(10,27)(11,21)(13,25)(14,24)(15,32)(16,30)(17,34)(20,29)(22,36)(23,31)(26,33)(28,35);(5,8)(6,13)(7,18)(9,17)(10,28)(11,24)(12,25)(14,21)(15,33)(16,22)(19,34)(20,31)(23,29)(26,32)(27,35)(30,36);(5,9,10,25,11,26,29,36)(6,14,15,31,16,18,34,35)(7,19,27,13,21,33,20,22)(8,17,28,12,24,32,23,30)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214887, #72265431, #72307310, …
maximal normal subgroup of	#72214887, #72265431, #72307310, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261672);