#203599

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

764 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)(6,9)(7,11)(10,12)(13,14,17,22,18,23,27,21)(15,19,16,25,24,20,26,28);(2,4)(6,10)(9,12)(13,15,17,16,18,24,27,26)(14,19,22,25,23,20,21,28);(5,6,7,9,8,10,11,12)(13,16,17,24,18,26,27,15)(14,20,21,25,23,19,22,28);(15,24)(16,26)(19,20)(25,28)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^b=di,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343947, #72343949, #72343963
maximal normal subgroup of	#72343947, #72343949, #72343963

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110795);