#369196

order	512
nilpotency class	3
isoclinism	isoclinic class of #25798
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3155	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1922 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281002, #72281024, #72281320, …
maximal normal subgroup of	#72281002, #72281024, #72281320, …
maximal quotient of	#72313525

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276392);