#354623

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle	C₂⁴	C₂³×C₄
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1932 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)(6,8)(13,14,19,21,18,17,16,25)(15,20,22,28,24,23,26,27);(2,4)(9,10,11,12)(13,15)(14,20)(16,26)(17,23)(18,24)(19,22)(21,28)(25,27);(5,6,7,8)(13,16,18,19)(14,21,17,25)(15,22,24,26)(20,27,23,28);(13,17,19,25,18,14,16,21)(15,23,22,27,24,20,26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264277, #72264306, #72264342, …
maximal normal subgroup of	#72264277, #72264306, #72264342, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261819);