#154039

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄²

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

3594 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)(5,6,7,8)(14,17)(15,19);(6,8)(9,10,12,11)(14,17)(18,20)(22,25)(26,28);(13,14)(15,18)(16,17)(19,20)(21,22,24,25)(23,26,27,28);(1,3,4,2)(5,6)(7,8)(9,11,12,10)(13,15,16,19)(14,18,17,20)(21,23,24,27)(22,26,25,28)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fg,d^2=eghi,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

derived subgroup of	#72267016, #72267022
Fitting subgroup of	#72267016
maximal subgroup of	#72266781, #72266782, #72266910, …
maximal normal subgroup of	#72266781, #72266782, #72266910, …
maximal quotient of	#72267697

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 61235);