#6394891

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#34963	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2409 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,d^b=dfi

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319163, #72319531, #72319551
maximal normal subgroup of	#72319163, #72319531, #72319551

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6302087);