#238090

order	512
nilpotency class	3
isoclinism	isoclinic class of #238090
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1463 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,d^c=di,e^c=eh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235251, #72235378, #72251343, …
maximal normal subgroup of	#72235251, #72235378, #72251343, …
maximal quotient of	#72255677

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145286);