#299104

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1404 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 20 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214806
maximal normal subgroup of	#72214806
maximal quotient of	#72214497

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206300);