#392618

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle	C₂⁴	C₂³×C₄
prime radical	C₂×#2388	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,h^2=i,b^a=be,c^a=cf,d^a=dg,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239380, #72239394, #72239395, …
maximal normal subgroup of	#72239380, #72239394, #72239395, …
maximal quotient of	#72286602, #72286813

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299814);