#228194

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2358
prime radical	#3475	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

697 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=deh,e^b=eh,d^c=di,f^c=fi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216030
maximal normal subgroup of	#72216030
maximal quotient of	#72216007, #72216052

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 135390);