#6394682

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	#34957	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2408 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)(7,11)(8,10)(9,12)(13,14,18,21,17,20,23,27)(15,22,24,28,16,19,25,26);(5,7)(6,9)(8,12)(10,11)(13,15,18,24,17,16,23,25)(14,19,21,26,20,22,27,28);(2,4)(13,16,18,25,17,15,23,24)(14,19,21,26,20,22,27,28);(7,12)(9,11);(6,10)(9,11)(14,20)(19,22)(21,27)(26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319551
maximal normal subgroup of	#72319551

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6301878);