#185473

order	512
nilpotency class	3
isoclinism	isoclinic class of #185463
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3518	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2253 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(2,6,7,8)(9,10)(11,19,22,15)(12,23,25,16)(13,17)(14,18)(20,28)(21,31,26,29)(24,36,38,30)(27,32)(33,35)(34,39)(37,40);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,20)(13,26)(14,22)(16,28)(17,31)(18,19)(21,27)(23,33)(24,37)(25,35)(29,32)(30,40)(34,38)(36,39);(9,12,14,25)(10,16,18,23)(11,20,22,35)(13,24,27,38)(15,28,19,33)(17,30,32,36)(21,34,26,37)(29,39,31,40);(1,3,5,4)(2,6,7,8)(9,13,14,27)(10,17,18,32)(11,21,22,26)(12,24,25,38)(15,29,19,31)(16,30,23,36)(20,34,35,37)(28,39,33,40)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=hi,e^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,g^b=ghi,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239232
maximal normal subgroup of	#72239232
maximal quotient of	#72239246

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92669);