#381416

order	512
nilpotency class	3
isoclinism	isoclinic class of #30056
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

712 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)(7,17)(8,13)(9,14)(10,26)(11,16)(12,19)(15,25)(18,33)(20,32)(21,28)(22,36)(23,30)(24,27)(29,34)(31,35);(5,7,10,19,11,20,25,9)(6,12,15,17,16,14,26,32)(8,21,22,24,23,18,34,35)(13,27,29,28,30,31,36,33);(5,8)(6,13)(7,18,20,21)(9,24,19,35)(10,22)(11,23)(12,27,14,31)(15,29)(16,30)(17,28,32,33)(25,34)(26,36);(5,9,25,20,11,19,10,7)(6,14,15,32,16,12,26,17)(8,21,22,24,23,18,34,35)(13,28,36,27,30,33,29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224750, #72224755, #72224758
maximal normal subgroup of	#72224750, #72224755, #72224758

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288612);