#359561

order	512
nilpotency class	3
isoclinism	isoclinic class of #23897
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3487	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3912 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)(3,4)(5,6)(7,12)(8,19,21,26)(9,14)(10,15)(11,16)(13,17,28,23)(18,25)(20,33,34,32)(22,29)(24,30)(27,35,36,31);(2,4)(5,7,11,10)(6,12,16,15)(8,17,21,23)(9,18,22,24)(13,26,28,19)(14,25,29,30)(20,31,34,35)(27,33,36,32);(5,8)(6,13)(7,17)(9,20)(10,23)(11,21)(12,19)(14,27)(15,26)(16,28)(18,31)(22,34)(24,35)(25,32)(29,36)(30,33);(2,4)(5,9,11,22)(6,14,16,29)(7,18,10,24)(8,20,21,34)(12,25,15,30)(13,27,28,36)(17,31,23,35)(19,32,26,33)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235198, #72251709, #72251718, …
maximal normal subgroup of	#72235198, #72251709, #72251718, …
maximal quotient of	#72236022

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266757);