#277972

order	512
nilpotency class	3
isoclinism	isoclinic class of #277971
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3100	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

607 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=e,d^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,g^b=gi,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185168);