#204138

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂²×C₂²⋊C₄
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1264 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215872, #72216045, #72216052, …
maximal normal subgroup of	#72215872, #72216045, #72216052, …
maximal quotient of	#72321875

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111334);