#394274

order	512
nilpotency class	3
isoclinism	isoclinic class of #394273
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#3026	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2811 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6)(7,11)(8,12)(9,13)(10,14)(15,22)(16,23)(17,24)(18,30)(19,32)(20,26)(21,27)(25,33)(28,36)(29,35)(31,34);(1,2,3,4)(5,7,10,17)(6,11,14,24)(8,18)(9,16,21,29)(12,22)(13,23,27,35)(15,20)(19,31)(25,34)(26,30)(28,33)(32,36);(5,8)(6,12)(7,15)(9,19)(10,20)(11,22)(13,25)(14,26)(16,28)(17,18)(21,33)(23,34)(24,30)(27,32)(29,31)(35,36);(5,9)(6,13)(7,16)(8,19)(10,21)(11,23)(12,25)(14,27)(15,28)(17,29)(18,31)(20,33)(22,34)(24,35)(26,32)(30,36)

Polycyclic presentation:

2^9|b^2=ei,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,e^c=ei,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72241077, #72241081, #72251368, …
maximal normal subgroup of	#72241077, #72241081, #72251368, …
maximal quotient of	#72241191, #72252219, #72258957, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301470);