#218458

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
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	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	#2011	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

755 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 125654);