#353825

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=f,b^a=be,c^a=cf,e^a=ei,h^a=hi,e^b=ei,g^b=gi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257293, #72257313, #72257361, …
maximal normal subgroup of	#72257293, #72257313, #72257361, …
maximal quotient of	#72264520

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261021);