#6370184

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3286 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,g^2=i,b^a=bf,d^a=df,c^b=cfi,e^d=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277380);