#154020

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
commutator	C₂³	C₂²×C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,d^2=egh,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72299621, #72299635, #72299637
maximal normal subgroup of	#72299621, #72299635, #72299637
maximal quotient of	#72299914

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 61216);