#373949

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

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

746 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=h,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,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, 281145);