#232158

order	512
nilpotency class	3
isoclinism	isoclinic class of #232153
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#19747	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

477 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)(3,8,11,7)(4,12,14,16)(5,9)(6,10)(13,15)(17,18)(19,29,31,32)(20,24,30,26)(21,27)(22,25)(23,28);(1,3)(2,7)(4,13)(5,14)(6,11)(8,10)(9,12)(15,16)(17,19,23,31)(18,24,28,26)(20,22,30,21)(25,29,27,32);(1,4,13,3,6,14,5,11)(2,8,9,12,10,7,15,16)(18,25,28,27)(24,32,26,29);(2,9,10,15)(7,16,8,12)(17,20,21,19,23,30,22,31)(18,26,27,32,28,24,25,29)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=efh,d^2=e,e^2=i,f^2=hi,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cef,d^b=de,e^b=ei,f^b=fhi,g^b=ghi,g^c=gh,g^d=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72245124, #72286642
maximal normal subgroup of	#72245124, #72286642

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139354);