#110398

order	512
nilpotency class	3
isoclinism	isoclinic class of #7694
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	C₄×C₂²⋊C₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

718 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:

(2,4)(5,6,9,12)(7,11,8,10)(13,14,17,21,18,22,16,19)(15,20,25,27,23,28,24,26);(6,10)(7,8)(11,12)(14,19,22,21)(15,23)(20,27,28,26)(24,25);(1,2,3,4)(5,7,9,8)(6,10,12,11)(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=efh,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,c^b=ch,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17594);