#370466

order	512
nilpotency class	3
isoclinism	isoclinic class of #24985
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

724 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277662);