#379409

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

910 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6)(7,11)(8,10)(9,12)(13,14,17,22,19,24,18,23)(15,25,28,20,27,35,29,36)(16,30,31,38,33,37,32,21)(26,34,41,43,40,44,42,39);(1,2,3,4)(5,7,8,12)(6,9,10,11)(13,15,17,28,19,27,18,29)(14,20,23,25,24,36,22,35)(16,26,31,41,33,40,32,42)(21,34,37,39,38,44,30,43);(2,4)(13,16,18,32,19,33,17,31)(14,21,22,30,24,38,23,37)(15,26,29,42,27,40,28,41)(20,34,35,43,36,44,25,39);(6,10)(9,11)(14,22,24,23)(15,27)(20,25,36,35)(21,30,38,37)(26,40)(28,29)(34,39,44,43)(41,42)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=f,d^2=fh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di

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, 286605);