#354040

order	512
nilpotency class	3
isoclinism	isoclinic class of #1909
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2914 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267226, #72267229, #72312492, …
maximal normal subgroup of	#72267226, #72267229, #72312492, …
maximal quotient of	#72267259, #72267413

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261236);