#423607

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1918 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264385, #72264404, #72264480
maximal normal subgroup of	#72264385, #72264404, #72264480

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330803);