#347053

order	512
nilpotency class	3
isoclinism	isoclinic class of #347042
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#62824	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2473 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=h,b^a=be,c^a=cf,d^a=dg,f^b=fh,g^b=gi,e^c=eh,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249447, #72266832, #72266833, …
maximal normal subgroup of	#72249447, #72266832, #72266833, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254249);