#108775

order	512
nilpotency class	2
isoclinism	isoclinic class of #94871
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	C₂×#95
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,d^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,g^a=gh,c^b=ci,f^b=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15971);