#202642

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
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	C₂×C₄×D₄
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

928 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215867, #72320225, #72321282, …
maximal normal subgroup of	#72215867, #72320225, #72321282, …
maximal quotient of	#72320710, #72323053

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109838);