#6398630

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#2410	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2236 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)(7,11)(8,10)(9,12)(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28);(5,7)(6,9)(8,12)(10,11)(15,21)(18,24)(22,27)(25,28);(2,4);(7,12)(9,11);(6,10)(9,11)(13,15,16,22,17,21,23,27)(14,18,19,25,20,24,26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307510, #72307546, #72307628, …
maximal normal subgroup of	#72307510, #72307546, #72307628, …
maximal quotient of	#72343143

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6305826);