#386719

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂³×D₄	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=efg,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,d^c=di

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, 293915);