#193528

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	#34937	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1622 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^b=eh,g^b=gh,d^c=dhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286123, #72343941, #72343958
maximal normal subgroup of	#72286123, #72343941, #72343958
maximal quotient of	#72286291, #72286813, #72286976

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100724);