#227624

order	512
nilpotency class	3
isoclinism	isoclinic class of #20158
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2358
prime radical	#18480	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

653 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dei,e^b=eh,d^c=dh,f^c=fi,e^d=eh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287402, #72322005, #72322077
maximal normal subgroup of	#72287402, #72322005, #72322077

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 134820);