#181911

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂⁴
commutator	C₂²	C₂²×C₄×C₈
socle	C₂³	C₂²×C₄²
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,f^2=h,g^2=i,b^a=be,d^a=di,c^b=ci

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320669, #72320715, #72320783
maximal normal subgroup of	#72320669, #72320715, #72320783
maximal quotient of	#72321391

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89107);