#455961

order	512
nilpotency class	3
isoclinism	isoclinic class of #455826
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1135 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363157);