#108617

order	512
nilpotency class	3
isoclinism	isoclinic class of #8626
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#380
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

770 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,e^c=eh,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15813);