#210883

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
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	#1999	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

826 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)(7,10)(11,12)(13,14,17,21,18,22,24,20)(15,23,16,19,25,28,26,27);(5,6,10,12,8,11,7,9)(13,15,24,26,18,25,17,16)(14,19,21,28,22,27,20,23);(2,4)(15,25)(16,26)(19,27)(23,28);(13,16,17,25,18,26,24,15)(14,19,21,28,22,27,20,23)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=g,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320150
maximal normal subgroup of	#72320150
maximal quotient of	#72320199

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 118079);