#6373368

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2424	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2345 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,e^b=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307067, #72307082, #72307108, …
maximal normal subgroup of	#72307067, #72307082, #72307108, …
maximal quotient of	#72343156

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280564);