#245547

order	512
nilpotency class	3
isoclinism	isoclinic class of #245519
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#43893	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1419 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=hi,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,e^b=eh,e^c=ei,e^d=eh,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240201, #72240205, #72240291, …
maximal normal subgroup of	#72240201, #72240205, #72240291, …
maximal quotient of	#72240315

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152743);