#184112

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

864 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)(5,6,9,11,8,7,10,12)(13,14)(15,19)(16,20)(17,22)(18,21)(23,26)(24,27)(25,28);(2,4)(6,7)(11,12)(13,15,17,16,18,23,25,24)(14,19,22,20,21,26,28,27);(6,7)(11,12)(13,16,17,23,18,24,25,15)(14,20,22,26,21,27,28,19);(5,7,9,12,8,6,10,11)(14,21)(19,26)(20,27)(22,28)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=f,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=di,c^b=ci,d^b=dh,d^c=dh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343924, #72343930, #72343974
maximal normal subgroup of	#72343924, #72343930, #72343974

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91308);