#450632

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

839 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252649, #72256869, #72256881
maximal normal subgroup of	#72252649, #72256869, #72256881

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357828);