#354024

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle	C₂⁴	C₂³×C₄
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3590 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,g^2=i,b^a=be,c^a=cf,d^b=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215932, #72215970, #72216085, …
maximal normal subgroup of	#72215932, #72215970, #72216085, …
maximal quotient of	#72215865, #72215897, #72321595, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261220);