#367159

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1516 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=f,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,g^c=gi

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, 274355);