#183858

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂⁴
commutator, socle	C₂³	C₂²×C₄²
prime radical	C₂²×D₄	C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

892 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)(5,6,9,11,8,7,10,12)(13,14)(15,23)(16,20)(17,21)(18,22)(19,24)(25,26)(27,28);(2,4)(6,7)(11,12)(13,15,18,25,17,24,27,20)(14,19,22,16,21,23,28,26);(6,7)(11,12)(13,16,18,23,17,26,27,19)(14,20,22,15,21,25,28,24);(5,7,9,12,8,6,10,11)(14,21)(15,24)(20,25)(22,28)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343120, #72343129, #72343201, …
maximal normal subgroup of	#72343120, #72343129, #72343201, …
maximal quotient of	#72343463, #72343486

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91054);