#231227

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	#1999	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

665 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ef,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^b=dei,d^c=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, 138423);