#353393

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁶	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4930 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)(5,6,7,8)(13,14)(15,22)(16,20)(17,25)(18,21)(19,23)(24,27)(26,28);(9,10,11,12)(13,15)(14,19)(16,23)(17,26)(18,24)(20,27)(21,28)(22,25);(2,4)(13,16,18,17)(14,20,21,25)(15,23,24,26)(19,27,28,22);(14,21)(19,28)(20,25)(22,27)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317037, #72317039, #72317042
maximal normal subgroup of	#72317037, #72317039, #72317042
maximal quotient of	#72317102

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260589);