#243868

order	512
nilpotency class	3
isoclinism	isoclinic class of #232755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

764 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2,4,6,5,7,3,8)(9,10,13,16,14,17,12,22)(11,18,20,23,21,24,19,15);(1,2)(3,6)(4,8)(5,7)(9,11,12,20,14,21,13,19)(10,15,22,18,17,23,16,24);(1,2)(3,6)(4,8)(5,7)(15,23)(18,24);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,18,23,24)

Polycyclic presentation:

2^9|a^2=e,b^2=efh,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,e^c=ei,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238923, #72238925, #72238947
maximal normal subgroup of	#72238923, #72238925, #72238947

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151064);