#417767

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

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

3222 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)(9,10)(11,12)(13,14)(15,25)(16,23)(17,22)(18,21)(19,24)(20,26)(27,28);(5,6)(7,8)(13,15,18,26,19,17,16,27)(14,20,23,25,24,28,21,22);(2,4)(6,8)(13,16,19,18)(14,21,24,23)(15,26,17,27)(20,25,28,22);(10,12)(13,17,19,15)(14,22,24,25)(16,27,18,26)(20,23,28,21)

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=dei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315768, #72315809, #72316213
maximal normal subgroup of	#72315768, #72315809, #72316213

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324963);