#774690

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3788 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279803, #72280075, #72280180, …
maximal normal subgroup of	#72279803, #72280075, #72280180, …
maximal quotient of	C₂×#774690

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 681886);