#202646

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂×C₄×D₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

806 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,11,14)(7,15,18,19)(8,9,13,10)(12,16,20,17)(21,22,24,29,26,30,25,27)(23,31,33,28,32,35,34,36);(5,7,9,16,11,18,10,17)(6,12,8,19,14,20,13,15)(22,27,30,29)(23,32)(28,35,36,31)(33,34);(2,4)(5,8,10,6,11,13,9,14)(7,15,17,20,18,19,16,12)(21,23,25,34,26,32,24,33)(22,28,29,35,30,36,27,31);(5,9,11,10)(6,13,14,8)(7,16,18,17)(12,15,20,19)(21,24,26,25)(22,29,30,27)(23,33,32,34)(28,35,36,31)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ch,d^c=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286059, #72343872, #72343878
maximal normal subgroup of	#72286059, #72343872, #72343878

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109842);