#394736

order	512
nilpotency class	3
isoclinism	isoclinic class of #394177
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2006 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)(13,14,19,24)(15,21,26,27)(16,20,25,28)(17,23,18,22);(5,6)(7,8)(13,15,18,25,19,26,17,16)(14,20,23,27,24,28,22,21);(2,4)(6,8)(13,16,17,26,19,25,18,15)(14,21,22,28,24,27,23,20);(10,12)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311954, #72311966, #72311995
maximal normal subgroup of	#72311954, #72311966, #72311995

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301932);