#332401

order	512
nilpotency class	3
isoclinism	isoclinic class of #332327
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

626 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)(3,10)(4,7)(5,9)(6,13)(8,14)(11,16)(12,15)(17,18,22,26)(19,27,20,29)(21,31,32,25)(23,30,24,28);(1,3)(2,6)(4,12)(5,11)(7,10)(8,16)(9,15)(13,14);(3,11)(12,13)(17,19,22,20)(18,23,26,24)(21,30,32,28)(25,29,31,27);(2,7)(3,12)(4,14)(8,9)(10,16)(11,13)(17,20,22,19)(18,24,26,23)(21,28,32,30)(25,27,31,29)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238925, #72238934, #72238937
maximal normal subgroup of	#72238925, #72238934, #72238937

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239597);