#110971

order	512
nilpotency class	3
isoclinism	isoclinic class of #110954
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1021 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

(2,7)(3,8)(4,5)(9,10)(11,14,18,21)(12,15)(13,16)(17,23,24,20)(19,22);(1,2,5,3,6,8,4,7)(9,11)(10,14,16,21)(12,17)(13,18)(15,20,22,23)(19,24);(1,3,4,2,6,7,5,8)(9,12,13,19)(10,15,16,22)(11,17,18,24)(14,20,21,23)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=hi,e^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gh,d^b=di,e^b=ei,f^b=fi,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, 18167);