#449704

order	512
nilpotency class	3
isoclinism	isoclinic class of #417532
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	C₂⁵	C₂⁴
prime radical	#35348	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225699, #72225705, #72225706, …
maximal normal subgroup of	#72225699, #72225705, #72225706, …
maximal quotient of	#72284780, #72317431, C₂×#449704

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356900);